coordinate system

Let (u,v) be a coordinate system whose origin is the same as the one for the (x,y) system and which is obtained by rotating the (x,y) coordinate axes by 45 degrees. How would I prove these coordinate transformation formulas
x = (sqrt(2)/2)*(u-v) and y = (sqrt(2)/2)*(u+v)?
I would rather use geometric reasoning than linear algebra. I'm sure it has something to do with sin(45) or cos(45), but I just can't see how to transform it.

Let (u,v) be a coordinate system whose origin is the same as the one for the (x,y) system and which is obtained by rotating the (x,y) coordinate axes by 45 degrees. How would I prove these coordinate transformation formulas
x = (sqrt(2)/2)*(u-v) and y = (sqrt(2)/2)*(u+v)?
I would rather use geometric reasoning than linear algebra. I'm sure it has something to do with sin(45) or cos(45), but I just can't see how to transform it.

Draw the two pairs of cartesian (rectangular) axes having the same origin (0,0), such that (u,v) is rotated 45 degees from (x,y).

Mark a random point (x,0) on the positive x-axis. Draw the vertical line x = x, or project the (x,0) on the (u,v) axes.